| L(s) = 1 | + (0.784 + 0.905i)2-s + (6.23 + 4.00i)3-s + (0.934 − 6.49i)4-s + (2.07 − 4.54i)5-s + (1.26 + 8.79i)6-s + (8.12 + 2.38i)7-s + (14.6 − 9.43i)8-s + (11.6 + 25.4i)9-s + (5.75 − 1.68i)10-s + (13.9 − 16.1i)11-s + (31.8 − 36.7i)12-s + (−12.7 + 3.74i)13-s + (4.21 + 9.23i)14-s + (31.1 − 20.0i)15-s + (−30.3 − 8.89i)16-s + (3.21 + 22.3i)17-s + ⋯ |
| L(s) = 1 | + (0.277 + 0.320i)2-s + (1.20 + 0.771i)3-s + (0.116 − 0.812i)4-s + (0.185 − 0.406i)5-s + (0.0860 + 0.598i)6-s + (0.438 + 0.128i)7-s + (0.649 − 0.417i)8-s + (0.430 + 0.941i)9-s + (0.181 − 0.0533i)10-s + (0.382 − 0.441i)11-s + (0.766 − 0.884i)12-s + (−0.272 + 0.0799i)13-s + (0.0805 + 0.176i)14-s + (0.536 − 0.344i)15-s + (−0.473 − 0.139i)16-s + (0.0458 + 0.318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.84234 + 0.443475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.84234 + 0.443475i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.07 + 4.54i)T \) |
| 23 | \( 1 + (109. + 14.3i)T \) |
| good | 2 | \( 1 + (-0.784 - 0.905i)T + (-1.13 + 7.91i)T^{2} \) |
| 3 | \( 1 + (-6.23 - 4.00i)T + (11.2 + 24.5i)T^{2} \) |
| 7 | \( 1 + (-8.12 - 2.38i)T + (288. + 185. i)T^{2} \) |
| 11 | \( 1 + (-13.9 + 16.1i)T + (-189. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (12.7 - 3.74i)T + (1.84e3 - 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-3.21 - 22.3i)T + (-4.71e3 + 1.38e3i)T^{2} \) |
| 19 | \( 1 + (16.8 - 117. i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-39.3 - 273. i)T + (-2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-53.4 + 34.3i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (81.0 + 177. i)T + (-3.31e4 + 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-11.4 + 25.1i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-24.8 - 15.9i)T + (3.30e4 + 7.23e4i)T^{2} \) |
| 47 | \( 1 + 313.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-22.8 - 6.70i)T + (1.25e5 + 8.04e4i)T^{2} \) |
| 59 | \( 1 + (630. - 185. i)T + (1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-379. + 243. i)T + (9.42e4 - 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-187. - 216. i)T + (-4.28e4 + 2.97e5i)T^{2} \) |
| 71 | \( 1 + (462. + 533. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-112. + 784. i)T + (-3.73e5 - 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-254. + 74.7i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-167. - 365. i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-1.26e3 - 815. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (136. - 299. i)T + (-5.97e5 - 6.89e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.64050676075465831173950061772, −12.23717532257592046267448630128, −10.70674973102399314484124540965, −9.854527789619740706090083119555, −8.904680503401845024876378781599, −7.918078682602477205144535858419, −6.21181770312597080269128240910, −4.93977740622743804564781407982, −3.72109179904234241957900614886, −1.79135031696026705708514633735,
2.01562685224140303758394294204, 2.99899630230164391867461941914, 4.47241533386485960897914041420, 6.73859472670700056215255429899, 7.66184243752908981877913445407, 8.449894525182871111440121940118, 9.711437092084506643121389817959, 11.25870616240833839353635000844, 12.15413476319451951064088255185, 13.25791919891705288978421778582
